An update on spaces of embeddings

The Cerf-Morlet comparison theorem states that the group of diffeomorphisms of the compact n-ball which is the identity on the boundary is an (n+1)-fold loop space, whose (n+1)-fold de-looping is the homotopy-quotient of the group of the PL-homeomorphisms of R^n by the smooth homeomorphisms of R^n. In 2002 I observed that this Cerf-Morlet iterated loop space structure generalizes to an iterated loop space structure on the space of embeddings of R^j in R^n which are standard outside of a fixed ball and equipped with a trivialization of a tubular neighbourhood. This talk will be about the homotopy-type of these embedding spaces, viewed as iterated loop-spaces. I've been able to make the most progress in the case (j,n)=(1,3), which I will describe in detail. My most recent work on this has to do with a `realization problem' associated to the (j,n)=(1,3) case which boils down to a study of the symmetry groups of a certain family of hyperbolic links in S3 together with a family of natural representations of these symmetry groups.